In the related art, an automatic door closing hinge configured to rotate an opened door in a closing direction automatically using a restoring force of a coil spring and provided with a shock-absorbing function for absorbing a shock applied when the door is closed using a hydraulic cylinder is known and, in recent years, an automatic door closing hinge configured to absorb a shock with an air damper instead of the hydraulic cylinder is proposed. The automatic door closing hinge includes a type, for example, configured in such a manner that a piston is stored and arranged in a cylinder provided on one of a pair of vanes, and an operating rod which is fixed at an upper portion to the other vane is arranged within the cylinder, balls disposed so as to roll over a predetermined position of the piston and projecting from an inner periphery are engaged with a cam groove being formed on an outer periphery of a lower portion of the operating rod and having an inclined portion, the piston is moved forward and backward corresponding to the movement of the ball with respect to the inclined portion of the cam groove, and a shock is absorbed by an air cushion action in the cylinder by the returning operation of the piston when the door is closed (see JP-A-2002-303072 and JP-A-2005-113682).